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Danny Pak-Keung Chan

Apparently, this user prefers to keep an air of mystery about them.

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10 If $\int_a^bg(x)dx=0$, show that $\int_a^bf(x)g(x)dx=0.$ May 20 '19
10 Spivak's Calculus: chapter 2, problem 18(c) Aug 5 '20
8 $f_n \rightharpoonup f$, $g_n \to g$ in measure, $\|g_n\|_{L^\infty} \leq M$, then $f_n g_n \rightharpoonup fg$, Jun 30 '18
7 Proof: If $f : [a,b] \to [0,M]$ is Riemann integrable and has integral zero then $f(x) = 0$ at every continuity point $x$ of $f$. Jul 22 '17
7 Prove that $\binom{n}{0} ^2 + \binom{n}{1} ^2 + \binom{n}{2} ^2 + … + \binom{n}{n} ^2 = \binom{2n}{n}$. Feb 23 '20
7 If $f$ is measurable and $fg$ is in $L^1$ for all $g \in L^q$, must $f \in L^p$? May 12 '19
6 For a differentiable function $f$ show that $\{x:\limsup_{y\to x}|f'(y)|<\infty\} $ is open and dense in $\mathbb R$ Dec 14 '19
5 Convergence in measure - product Jul 7 '17
5 Conditional expectation for a sum of iid random variables: $E(\xi\mid\xi+\eta)=E(\eta\mid\xi+\eta)=\frac{\xi+\eta}{2}$ Aug 7 '17
4 If $\lim_{x\to\infty}f'(x)=0$ and if $g\geq c>0$ on $(a,\infty)$, then $\lim_{x\to\infty}\left[f\left(x+g(x)\right)-f(x)\right]=0.$ Feb 5
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